Complexity As a Homeomorphism Invariant for Tiling Spaces Tome 67, No 2 (2017), P

R AN IE N R A U L

I ’ I

N S ANNALES

DE L’INSTITUT FOURIER

Antoine JULIEN Complexity as a homeomorphism invariant for tiling spaces Tome 67, no 2 (2017), p. 539-577.

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COMPLEXITY AS A HOMEOMORPHISM INVARIANT FOR TILING SPACES

by Antoine JULIEN (*)

Abstract. — It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the complexity is polynomial, that the exponent of the leading term is preserved by homeomorphism). This theorem can be reworded in terms of d-dimensional d infinite words: if two Z -subshifts (with the same conditions as above) are flow equivalent, their complexity functions are equivalent up to rescaling. An analogous theorem is stated for the repetitivity function, which is a quantitative measure of the recurrence of orbits in the tiling space. Behind this result is the fact that any homeomorphism between tiling spaces is described by a so-called “shape deformation”. In the last section, we use this observation to show that a certain cohomology group is an invariant of homeomorphisms between tiling spaces up to topological conjugacy. Résumé. — Il est prouvé dans cet article que deux pavages apériodiques et répétitifs dont les espaces de pavages sont homéomorphes ont des fonctions de complexité asymptotiquement équivalentes en un certain sens. Cela implique que lorsque les fonctions de complexité croissent polynomialement, l’exposant du terme dominant est préservé par homéomorphisme. Ce théorème peut s’énoncer en termes d de mots infinis d-dimensionels: si deux sous-décalages indexés par Z (avec les mêmes conditions) sont « flot-équivalents » (c’est-à-dire que leurs suspensions sont homéomorphes), alors leurs fonctions de complexité sont équivalentes à changement d’échelle près. Un théorème analogue peut être énoncé pour la fonction de répé- titivité, qui donne une mesure quantitative de la vitesse de récurrence des orbites dans l’espace de pavages. De manière sous-jacente, se trouve le fait que tout ho- méomorphisme entre espaces de pavages est induit par une déformation des tuiles. Dans la dernière section, on utilise cette observation pour montrer qu’un certain groupe de cohomologie fournit un invariant des homéomorphismes entre espaces de pavages à conjugation près.

Keywords: aperiodic tilings, complexity, repetitivity, flow-equivalence, orbit-equivalence. Math. classification: 37B50, 37B10. (*) Part of the early work on this topic was done at the University of Victoria, under the supervision of Ian Putnam, partially funded by the Pacific Institute for the Mathematical Sciences. 540 Antoine JULIEN

1. Introduction

This article is concerned with the study of aperiodic tilings. These tilings, such as the well-known Penrose tilings, provide good models for quasicrystals in physics. The tilings in which we are interested display two a priori antagonistic properties: a highly ordered structure (in the form of uniform repetition of patches), and aperiodicity. A tiling is a covering of the plane by geometric shapes (tiles), with no holes and no overlap. In this paper, tilings are assumed to be made using finitely many tile types up to translation. Furthermore, it is assumed that any two tiles can be fitted together in only finitely many ways (for example it could be assumed that tiles meet full-face to full-face in the case of polytopal tiles). This last condition is known as finite local complexity. Finally, tilings we are dealing with are assumed to be repetitive in the sense that any finite patch of a given tiling repeats within bounded distance of any point in the tiling (these definitions are given with more precision and quantifiers in Section2). An aperiodic tiling with the above properties has a topological compact space associated with it. It consists of all tilings which are indistinguish- able from it at a local scale, and it supports an Rd-action (for tilings of dimension d), induced by translations. It is possible to gain a better understanding of such tiling spaces by using an analogy with subshifts: a d multi-dimensional infinite word w ∈ AZ has a subshift associated with it. d It is the closure in AZ of the Zd-orbit of w under the shift (or translation) map. The word w can also be interpreted as a tiling by cubes, the colours of which are indexed by A. The tiling space of this tiling by cubes then corresponds exactly to the suspension of the subshift (which for d = 1 is also known as the mapping torus of the shift map). This suspension contains the subshift as a closed subset, and supports an Rd-action which extends the Zd-action on the subshift. We see here that if a subshift is minimal (which corresponds exactly to the condition cited above on repetition of patches), all elements in the subshifts have the same language, i.e. the same set of finite subwords. In the same way, if a tiling space is minimal, it make more sense to study the space rather than to particularise an arbitrarily chosen tiling. This philosophy of studying the space rather than the tiling (or the subshift rather than the word) is far from new, and has been quite successful. One can specifically cite the gap-labeling theorems, which relate the topological invariants of the space on the one hand (the ordered K0-group), and the gaps in the spectrum of a certain Schrödinger operator with aperiodic

ANNALES DE L’INSTITUT FOURIER COMPLEXITY AND HOMEOMORPHISMS BETWEEN TILING SPACES 541 potential on the other hand. Any attempt to give complete references for this problem would be unfair; we will therefore give a very incomplete list of references in the form of the papers [4, 13,3] and their references. A natural question, however, which results directly from this “space-over- tiling” approach is the following: whenever a relation is established between two tiling spaces, to what extent does it translate back in terms of tilings? In this paper, we start with two tiling spaces which are equivalent in the realm of topological spaces, i.e. homeomorphic. We then investigate what are the consequences on the underlying tilings. The main results state that whenever two tiling spaces are homeomorphic, their respective complexity functions, as well as their repetitivity functions are equivalent in some sense. Given a tiling (or a word), the complexity function is a map r 7→ p(r), where p(r) counts the number of distinct patches of size r, up to translation. This function was first studied in the framework of symbolic dynamics (one- dimensional subshifts). In their seminal paper, Morse and Hedlund [20] define p(n) as the number of subwords of size n in a given bi-infinite word. This function provides a good measure of order and disorder in a word: if p(n) doesn’t grow at least linearly, then the word is periodic. This result was generalized outside of the symbolic setting, in higher dimension: if the complexity function associated with a tiling doesn’t grow at least linarly, then the tiling is completely periodic i.e. it has d independent periods. One can refer to Lagarias–Pleasants [18] for a proof of this result in the setting of Delaunay sets of Rd. The first main theorem of this article describes how the complexity function is preserved whenever two tiling spaces are homeomorphic.

Theorem (Theorem 3.1).— If h :Ω −→ Ω0 is a homeomorphism between two tiling spaces of aperiodic and repetitive tilings with ﬁnite local complexity, then the associated complexity functions p and p0 satisfy the following inequalities:

0 cp(mr) 6 p (r) 6 Cp(Mr), for some constants c, C, m, M > 0 and all r big enough.

A few remarks: (1) If the functions p and p0 are at least linear (which is the case for non-periodic tilings, by the Morse–Hedlund theorem), then c and C can be taken equal to 1 (up to adjusting the values of m and M); (2) On the other hand, if the functions p and p0 grow at most polynomially, then m and M can be chosen equal to 1 (up to adjusting c

TOME 67 (2017), FASCICULE 2 542 Antoine JULIEN

and C). In particular, if p grows like a polynomial, then so does p0, and the exponent of the leading term is the same.

Corollary 1.1. — Let T be an aperiodic, repetitive tiling with ﬁnite local complexity. Assume that its complexity function grows asymptotically like rα, up to a multiplicative constant. Then the exponent α is a topological invariant of the tiling space associated with T .

The repetitivity function was also introduced by Morse and Hedlund [20] in the setting of one-dimensional symbolic dynamics. The repetitivity function for a tiling T is defined as follows: for r > 0, R(r) is the infimum of all numbers c which satisfy that any patch of radius c in T contains a copy of all patches of radius r which appear somewhere in T . It is well defined and finite for all r exactly when the tiling T is repetitive i.e. satisfies the afore- mentioned condition on repetition of patches. It is a quantitative measure of the patch-repetition property. It can also be seen as a measure of order and disorder in a tiling dynamical system, which is distinct from the complexity, though related, and is sometimes finer. For example, if R(r) doesn’t grow at least linearly, the tiling has d independent periods [18]. Low repetitivity implies low complexity: Lenz [19] proved that if R is bounded above by a linear function, then the complexity function of the tiling is bounded above by Crd for some constant C. On the other hand, it is possible to build some tilings which have low complexity (sub-linear in dimension 1), but have a repetitivity function which grows much faster than linearly (a Sturmian sequence with Lebesgue-typical parameter will provide such an example [20]).

Theorem (Theorem 3.5).— If h :Ω −→ Ω0 is a homeomorphism between two tiling spaces of aperiodic and repetitive tilings with ﬁnite local complexity, then the associated repetitivity functions R and R0 satisfy

0 cR(mr) 6 R (r) 6 CR(Mr) for some constants c, C, m, M > 0 and all r big enough.

In the literature, it is more common to make the qualitative distinction between linearly repetitive tiling spaces and those which are not. A tiling space is linearly repetitive (or linearly recurrent) if its repetitivity function satisﬁes R(r) 6 λr for some constant λ. Linearly repetitive tiling spaces enjoy nice properties (for example they are uniquely ergodic), and many of the actual models for quasicrystals are linearly repetitive. This property is preserved by homeomorphism

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Corollary. — If two tiling spaces satisfying the assumption of the previous theorem are homeomorphic, then either they are both linearly repetitive, or neither is. The results above show that there exists a significant relationship between the topology and the dynamics of a tiling space on the one hand, and its complexity and repetitivity function on the other hand. This is a priori surprising, since the complexity and repetitivity do not stand out naturally as being topological objects. The repetitivity function clearly is strongly related to the dynamics on the tiling space, as it can be interpreted in terms of maximal size of return vectors on a transversal, or as a measure of the recurrence of orbits in the dynamical system. The complexity function can be also related with the dynamics of the action of Rd on the tiling space. Given a tiling space, it is possible to define the patch-counting entropy (or configurational entropy) by the formula: log p(r) Hpc(Ω) = lim sup d . r→+∞ r It was proved to be equal to the topological entropy of (Ω, Rd) when Ω is a tiling space with finite local complexity [1]. In particular, using the results above, it is possible to state the following result.

Proposition. — Let Ω, Ω0 be two aperiodic, repetitive, FLC tiling spaces which are homeomorphic. Then (Ω, Rd) has positive topological entropy if and only if (Ω0, Rd) does. Note that when the entropy is not zero, it does not need to be preserved. When the entropy is zero, however, the complexity function appears rather to be a metric object (when the space is endowed with a commonly used distance, sometimes called the combinatorial metric). It had been no- ticed before [12,9], and it is discussed in more details in Section 3.1. The consequences of the results proved in this paper can be summed up by the following statement.

Proposition (Corollary 3.4).— A homeomorphism between two aperiodic, repetitive, FLC tiling spaces preserves the box-counting dimension of their transversals, when they are endowed with the combinatorial distance. The relationships between complexity on the one hand, and the dynamics on the other hand, had already been investigated to some extent. For example, for subshifts, it is known that a topological conjugacy is a so- called “sliding block code” (which is referred to as “local derivation” in our context, see Deﬁnition 2.3), and asymptotic properties of the complexity

TOME 67 (2017), FASCICULE 2 544 Antoine JULIEN function are unchanged by an invertible recoding. For tiling spaces, topological conjugacy and local derivation are however not synonymous. Frank and Sadun [8] proved in a more general setting (including even inﬁnite local complexity), that the asymptotic behaviour of the complexity function is preserved by a topological conjugacy of the tiling spaces. The reason why a mere homeomorphism (and not a conjugacy) has consequences beyond the topology has to do with the special structure of the space. Under the hypotheses made in this paper, a homeomorphism automatically sends orbits to orbits, in a rather controlled way.

Proposition (Theorem 3.10 and Corollary 5.2).— A homeomorphism h between two aperiodic, repetitive tiling spaces with finite local complexity is an orbit equivalence. Furthermore, for all T , the cocycle hT defined by d h(T − x) = h(T ) − hT (x) is a homeomorphism from R onto itself, and satisfies the following inequality:

khT (x)k 6 M kxk + C for some constants M,C which do not depend on T .

This result, together with an approximation lemma (Lemma 4.1), is the key ingredient to extend Frank and Sadun’s ideas and prove the main results above. The methods for proving Theorem 3.1 (namely Lemma 5.1) can also be related with results presented in [9, Chapitre 6]. It was shown that the image of a tiling by a small deformation has the same complexity function as the original tiling (up to equivalence). A small deformation induces a special kind of homeomorphism between the original tiling space and the space of the deformed tiling. The result of [9] can therefore be seen as a particular situation of the case treated in the present paper. The result of [9] was in particular applied to a family of deformations introduced by Sadun and Williams. In [24], it is proved that a tiling space is homeomorphic to the suspension of a subshift; in [9], it is proved that this homeomorphism preserves the complexity. It turns out not to be a coincidence that methods which applied for the study of complexity of deformed tilings can be used here. In a last section, we explain some of the links between homeomorphisms and deformations. Given h :Ω → Ω0, there is an underlying deformation (in the sense of Clark and Sadun) attached to it. This deformation is deﬁned by approximation, and some choices are made. However, given two choices, the deformations associated with h diﬀer by an asymptotically negligible element. Therefore,

ANNALES DE L’INSTITUT FOURIER COMPLEXITY AND HOMEOMORPHISMS BETWEEN TILING SPACES 545 such a homeomorphism h deﬁnes an element in the “mixed” cohomology 1 d group Hm(Ω, R ) deﬁned by Kellendonk [15], and the following result holds.

Theorem (Theorem 6.1).— Let hi :Ω → Ωi (i = 1, 2) be two homeomorphisms between aperiodic, repetitive, FLC tiling spaces. They each 1 d deﬁne a cohomology class [hi] ∈ Hm(Ω; R ). If [h1] = [h2], then Ω1 and Ω2 are topologically conjugate.

This mixed group describes homeomorphisms between tiling spaces, up to topological conjugacy. A more thorough investigation of the matters raised in the last section of this paper are the subject of a later paper [11].

Acknowledgements

I wish to thank Ian Putnam, Johannes Kellendonk, Lorenzo Sadun and Christian Skau for useful discussions. I also wish to thank Michael Baake who gave me the opportunity to present an early version of this work at the University of Bielefeld.

2. Preliminaries

2.1. Tilings and tiling spaces

In our discussion, a tile is a closed compact set of the Euclidean space Rd, which is homeomorphic to a closed ball, and is the closure of its interior. A prototile is the equivalence class under translation of a tile. Note that a tile may be decorated (one can think of blue and red squares, for example). Formally, it would be a pair (t, l), where t is the tile itself (a subset of Rd) and l is a label. This notation will remain implicit, and a tile will just be denoted by t. Let A be a finite collection of prototiles, which is fixed from this point on. A tiling with prototiles in A is a collection of tiles T = {ti}i∈I , such that for all i, the translation class of ti belongs to A. Furthermore, the union of the tiles is all of Rd, and the tiles may only intersect on their boundaries: ti ∩ tj has no interior, unless i = j. Given a tile t ⊂ Rd, and x ∈ Rd, the tile t − x is defined as the translate of t by x. In a straightforward way, T − x is defined as {ti − x}i∈I . It is a

TOME 67 (2017), FASCICULE 2 546 Antoine JULIEN tiling, and is in general diﬀerent from T (in our discussion, tilings are not being identiﬁed when they are translates of each other). A tiling T is periodic if there exists x ∈ Rd \{0}, such that T − x = T . It is called non-periodic or aperiodic otherwise. Given a non-periodic tiling, it makes sense to study the set of all tilings which cannot be distinguished from it at the local scale. This is the mo- tivation for constructing the tiling space, in the same way that a subshift can be associated with a word.

Definition 2.1. — Given T a tiling of Rd, made from a set of prototiles A, the hull of T is the following closure:

d ΩT = {T − x ; x ∈ R }.

The closure is taken in the set Ωall of all tilings made from tiles in A, for the topology of the distance deﬁned by:

d D(T1,T2) 6 ε if ∃ x1, x2 ∈ R , with kxik 6 ε,

such that (T1 − x1) agrees with (T2 − x2) on B(0, 1/ε), √ and D is bounded above by 1/ 2.

Note that all elements in ΩT are tilings themselves, and therefore there d is an action of R by translation on ΩT . This action is continuous for the topology defined above. A tiling T is said to have finite local complexity (or FLC) if for all R > 0 there is a finite number (up to translation) of patches of the form T ∩ B(x, R), for x ∈ Rd. Here, T ∩ B(x, R) stands for the patch of all tiles which intersect the ball B(x, R). A tiling T is said to be repetitive if for all finite patch P ⊂ T , there exists R > 0 such that a translate of P appears in T ∩ B(x, R), for any x ∈ Rd. The following proposition is well known.

Proposition 2.2. — Let T be a non-periodic and repetitive tiling, with finite local complexity. Then the tiling space ΩT is a compact space, and the action by translation is minimal. When there is no risk of ambiguity, the tiling space is just written Ω. Note that all these definitions (which hold for tilings) will also hold for other patterns, as long as it is possible to define what a patch is, and what “identical up to translation” means. Delaunay sets are an important example. A Delaunay set of Rd is a subset X ⊂ Rd, which is uniformly discrete and relatively dense. These two conditions mean respectively that

ANNALES DE L’INSTITUT FOURIER COMPLEXITY AND HOMEOMORPHISMS BETWEEN TILING SPACES 547 there is a constant m such that any two points in X are separated by a distance at least m; and a constant M such that any point in Rd is within distance at most M of a point in X. The definition of patch (or local configuration) of X now becomes transparent when using the notation X ∩ B(x, r). From there, the definitions of FLC, repetitivity and aperiodicity, as well as the definition of the hull (and its properties) are unchanged.

2.2. Local derivations, and transversals

Given a tiling T , the associated tiling space ΩT is sometimes called the “continuous” hull. It is also possible to describe a “discrete” tiling space, which is the analogue of a subshift in the symbolic setting (see Section 2.5). Amongst other properties, it is a subset of Ω, it intersects all Rd-orbit and its intersection with any orbit is countable. It does not carry a Zd- action in general. This discrete tiling space is sometimes called a canonical transversal. The canonicity of this transversal, however, leaves a lot of room for choice. First we deﬁne what a local derivation is.

Definition 2.3 (see [2]).— Let Ω and Ω0 be tiling spaces of repetitive and FLC tilings. A local derivation is a factor map (i.e. it is onto and commutes with translations) ϕ :Ω → Ω0 which satisﬁes the following local condition:

∃ r, R > 0, ∀ T ∈ Ω, ϕ(T ) ∩ B(0, r) only depends on T ∩ B(0,R).

In particular, for all x, ϕ(T ) ∩ B(x, r) only depends on T ∩ B(x, R). These conditions really mean that ϕ is a map deﬁned on patches of size R, and ϕ(T ) is obtained by gluing together the images under ϕ of the patches out of which T is made. In this case, ϕ(T ) is said to be locally derived from T . When ϕ is invertible and ϕ−1 is also a local derivation, T and ϕ(T ) are called mutually locally derivable, or MLD. Such maps are the analogues of sliding block codes in symbolic dynamics.

Definition 2.4. — Let Ω be a tiling space, and D be a local derivation rule deﬁned on Ω, such that for all T ∈ Ω, D(T ) is a Delaunay set. Such a rule is called a local pointing rule. The canonical transversal associated with D is the set

ΞD = {T ∈ Ω ; 0 ∈ D(T )}.

TOME 67 (2017), FASCICULE 2 548 Antoine JULIEN

Figure 2.1. A local pointing rule, as deﬁned on a set of rectangular tiles of size 2 × 1. Given any domino tiling made from these tiles (with dominoes meeting full-edge to full-edge), its associated point pattern will be periodic, even when the tiling is not.

The Delaunay set D(T ) is repetitive and has ﬁnite local complexity, since T has these properties itself. It does not need to be MLD to T , or even aperiodic (see Figure 2.1). In the literature, “the canonical transversal” is usually built by choosing a pointing rule D which selects exactly one point in the interior of each prototile (for example their barycentre if the tiles are convex), and by pointing the tilings consistently. Canonical transversals in the sense above are abstract transversals in the sense of Muhly Renault and Williams [21]. The reason why they are called “canonical” in spite of their apparent lack of canonicity is because T and T 0 are close in a canonical transversal if they agree exactly on a large ball centred at the origin (not up to a small translation). Remark 2.5. — If Ξ is a canonical transversal in a tiling space Ω, the topology on Ξ is induced by the family of clopen sets:

UP = {T ∈ Ξ; P ⊂ T }, where P is in the family of admissible patches. Equivalently, it is induced by the metric: √ D0(T,T 0) = inf ε > 0 ; T ∩ B(0, ε−1) = T 0 ∩ B(0, ε−1) ∪ 1/ 2 .

Another description of the “canonical” transversals is that they are vertical with respect to a (truly) canonical solenoid structure on the tiling space.

Theorem 2.6 (see for ex. [3,5]) .— The tiling space associated with a repetitive, aperiodic and FLC tiling is an abstract (so-called ﬂat) solenoid in the following sense: for each T ∈ Ω, there is a neighbourhood V of T , and a chart map ϕ, which maps V homeomorphically to the direct product

ANNALES DE L’INSTITUT FOURIER COMPLEXITY AND HOMEOMORPHISMS BETWEEN TILING SPACES 549 of a ball of Rd by a Cantor set. Furthermore, ϕ commutes with the action of Rd by translation whenever this action is deﬁned. Also, the transition maps satisfy the property:

−1 0 (2.1) ψ ◦ ϕ (x, ξ) = (x − tψ◦ϕ−1 , ξ ),

0 where ξ depends continuously on ξ, but the translation vector tψ◦ϕ−1 only depends on the chart maps.

Proof. — This is a partial proof in order to outline how the canonical solenoid structure is chosen. Property (2.1) will not be proved here. Let T ∈ Ω. Let P = T ∩ B(0, 1) be a small patch of T around the origin, and let ε > 0 be small with respect to the inner radius of the tiles. Deﬁne the derivation rule

0 d 0 D(T ) = {x ∈ R ; P ⊂ T − x}.

It is clearly a local rule, and the associated transversal ΞD is the set of all tilings which coincide with T up to radius at least 1. If T is repetitive, aperiodic and has ﬁnite local complexity, it is well known that ΞD is a Cantor set, and that D(T 0) is a Delaunay set for all T 0. Now, consider the map

ϕ : B(0, ε) × ΞD −→ Ω (x, T 0) 7−→ T 0 − x. It is clearly continuous, by continuity of the action. It is one-to-one, because ε was chosen small. It is therefore a homeomorphism onto its image, by compactness of the source space. Finally, it is an exercise to check that the image of ϕ restricted to B(0, ε) × ΞD is open in Ω. It provides the desired chart. It is important to remark that, by construction, the chart d maps commute with the action of R whenever it is defined. This motivates the following definition of vertical transversal. The fact that it coincides with the previously defined notion of canonical transversal comes from the definition of the solenoid structure and is left to the reader to check.

Definition 2.7. — The canonical transversals in the sense of Deﬁni- tion 2.4 are exactly the vertical transversals for this solenoid structure, that is: subsets Ξ ⊂ Ω which can be covered by chart boxes in such a way that for each such box V ' B(0, εV ) × XV ,

Ξ ∩ V '{0} × XV .

TOME 67 (2017), FASCICULE 2 550 Antoine JULIEN

2.3. Complexity

The complexity function of a tiling T is a function p, such that p(r) counts the number of patches of size r in T . This requires to define what “patch of size r” means in this context. While, in the 1-dimensional symbolic setting, there is no question about what a word of length n is, the situation is not so obvious in the continuous case. However, it appears that the asymptotic properties of the complexity function are unchanged by picking any reasonable definition of “patch of size r”. First, remark that a tiling T needs to have FLC in order to have a well defined complexity function (at least, in terms of naively counting patches, see [8] for ways of defining complexity for tilings without FLC). Second, if a tiling T is repetitive, then all tilings in its hull have the same complexity function. Therefore, for minimal tiling spaces, a complexity function is associated with the space rather than with one specific tiling. This justifies the following definition.

Definition 2.8. — Let Ω be a space of tilings which are repetitive, FLC and aperiodic. Let D be a local pointing rule, and Ξ the associated transversal. Then, the complexity function associated with these data, noted pΞ or pD is: pΞ(r) = Card T ∩ B(0, r); T ∈ Ξ = Card (T0 − x) ∩ B(0, r); x ∈ D(T0) for any T0 ∈ Ω. According to this deﬁnition, the complexity function counts the number of pointed patches of size n, where the pointing has to be chosen in advance and is not canonical to the tiling. However, it turns out that changing the transversal doesn’t change the complexity function.

Proposition 2.9. — Let Ω be a tiling space, and Ξ1, Ξ2 be two canonical transversals associated with pointing rules D1 and D2 respectively. Then the complexity functions pΞ1 and pΞ2 are equivalent in the following sense:

λ1pΞ2 (r − c) 6 pΞ1 (r) 6 λ2pΞ2 (r + C) for some constants c, C, λ1, λ2 > 0 and all r big enough. Proof. — Let T ∈ Ω. First, let D be the pointing rule deﬁned by

D(T ) = D1(T ) ∪ D2(T ). It is a pointing rule itself: D(T ) is locally derived from T , it is relatively dense, and the uniform discreteness comes from ﬁnite local complexity of T (and hence, ﬁnite local complexity of D(T )).

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Let Ξ be the transversal defined by D. It is sufficient for proving the theorem to show that pΞ is equivalent (in the sense above) to pΞi , for i = 1, 2. Let us prove it for i = 1. For any given r, let Xr be the set of all patches of the form (T − x) ∩ B(0, r), with x ∈ D(T ). Let Yr be the similarly defined set with x ∈ D1(T ).

It is obvious that pΞ1 (r) 6 pΞ(r) for all r (since Yr ⊂ Xr). Conversely, let R be the constant such that for all x ∈ Rd, B(x, R) intersects D1(T ). Let λ > 0 be the maximum number of points in D(T )∩B(x, R), for x ∈ Rd. It is well defined and finite (because of volume considerations using uniform discreteness of D(T ), for example). For each patch P ∈ Xr, 0 0 0 there is a patch P ∈ Yr+R such that P is a subpatch of P ; this patch P can contain at most λ distinct translates of patches in Xr. This defines a map Xr → Yr+R; each element in Yr+R has at most λ preimages under this map. Therefore,

pΞ(r) 6 λpΞ1 (r + C).

Patch of radius R + r

(T ) ∈ D1 R Patch of radius r ≤ (T ) ∈ D

Figure 2.2. D1 is sparser than D; yet, any patch of size r centered at a point of D is a subpatch of some patch of size R + r centred at a point of D1.

Since the choice of a transversal does not change the asymptotic behaviour of the complexity function, it is acceptable to write “the complexity function of T ” when there is no risk of ambiguity. It is understood that an unspeciﬁed transversal is chosen, the choice of which is not important.

Remark 2.10. — It can also be proved that these pointed complexity functions pΞ are equivalent to an unpointed complexity function p, where p(r) is the number of equivalence classes up to translation of patches of the form T ∩ B(0, r), for T ∈ Ω. It is proved in [9, Proposition 1.1.20].

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Changing the transversal does not change the complexity. It is worth noticing that applying an invertible local derivation to a tiling also preserves the complexity function. It can be proved directly by remarking that a local derivation is essentially a map from patches to patches. The proof is not included here. See Theorem 3.13 for a more general result. Proposition 2.11. — Let T ∈ Ω and T 0 ∈ Ω0 be two tilings which are mutually locally derivable. Let D(T ) be a Delaunay set derived from T (and hence locally derived from T 0 by mutual local derivability of T and T 0). Call Ξ ⊂ Ω and Ξ0 ⊂ Ω0 the canonical transversals relative to these Delaunay sets. Then their complexity functions satisfy:

pΞ(r − c) 6 pΞ0 (r) 6 pΞ(r + C), for r big enough. The equivalences between complexity functions in the propositions above are quite ﬁne. It is appropriate for our purpose to have a coarser notion of equivalence.

Definition 2.12. — Two complexity functions p1 and p2 are equivalent if there are C1,C2, m, M > 0 such that for all r big enough:

C1p2(mr) 6 p1(r) 6 C2p2(Mr). Remark that it is indeed a coarser notion than the ones above. However, the notion of a “patch of size r” is itself not uniquely defined: it depends on a choice of norm of Rd. By equivalence of norms in finite dimension, two different choices of norms would produce equivalent complexity functions in the sense above. Therefore, and unless there is a reason to prefer one norm over others, it is fine to use the coarser version.

2.4. Repetitivity The repetitivity of a tiling measures how often patches of a given size repeat. It was introduced in the setting of symbolic dynamics by Hedlund and Morse [20]. In a more general setting, it has been studied by Lagarias and Pleasants for Delaunay sets [18]. Definition 2.13. — Let Ω be a repetitive, aperiodic and FLC tiling space, and D be a pointing rule deﬁning a transversal Ξ. The repetitivity function is a function r 7→ RD(r) or RΞ(r) deﬁned by

RΞ(r) = inf{R > 0 ; ∀ T,T0 ∈ Ξ, ∃ x ∈ B(0,R),

T − x ∩ B(0, r) = T0 ∩ B(0, r)}.

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In words: RΞ(r) < C means that for any pointed patch of size r of the form T0 ∩ B(0, r) and for any T ∈ Ξ, that patch occurs in T within distance at most C of the origin (i.e. T −x and T0 agree up to distance r with kxk 6 C). The repetitivity function can be interpreted in dynamical terms. Let Ω be a tiling space with canonical transversal Ξ. Then for each r there is a partition Pr of Ξ in clopen sets of the form

0 0 ΞT,r := {T ∈ Ξ; T ∩ B(0, r) = T ∩ B(0, r)}.

Then RΞ(r) measures is an upper bound on the lengths of the shortest return vectors from any point in Ξ to any element of the partition Pr. Once again, it can be proved that the repetitivity function of a tiling space is independent of the choice of the transversal, modulo constants and rescaling.

Proposition 2.14. — Let Ωbe an aperiodic repetitive FLC tiling space, and Ξ1, Ξ2 be two canonical transversals corresponding to pointing rules D1 and D2 respectively. Then

RΞ2 (r − c) − K 6 RΞ1 (r) 6 RΞ2 (r + c) + K for some constants c, K > 0 and all r big enough. Lagarias and Pleasants [18] showed that for aperiodic repetitive tilings, the repetitivity function grows at least linearly. It is therefore harmless to drop the constant K in the theorem above, up to adjusting c. Proof. — As in the proof of the analogous result for complexity (Propo- sition 2.9), we may assume without loss of generality that Ξ1 ⊂ Ξ2.

It is straightforward that RΞ1 (r) 6 RΞ2 (r). Conversely, let R0 be the a radius (obtained by relative density) such that for any T ∈ Ξ2, there exists x ∈ B(0,R0) such that T − x ∈ Ξ1. Let now r > 0 and T,T0 ∈ Ξ2 be given. d Let us show that there is a y ∈ R with kyk 6 2R0 + RΞ1 (r + R0) + ε for arbitrarily small ε such that (T − y) ∩ B(0, r) = (T0 − y) ∩ B(0, r). Let x and x0 be of norm at most R0 such that T − x and T0 − x0 are in Ξ1. Next let y0 be a vector such that (T − x) − y0 and T0 − x0 agree up to radius r + R0. By deﬁnition of the repetitivity function, it can be chosen of norm at most RΞ1 (r +R0)+ε for any ε > 0. Now, it means that (T −x−y0)+x0 and T0 agree up to radius r, since kx0k 6 R0. So if y = y0 + x − x0, then T − y and T0 agree up to radius r. We have control on the norm of y, so that:

RΞ2 (r) 6 RΞ1 (r + R0) + 2R0.

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2.5. Formalism of symbolic dynamics

Tiling spaces also appear as suspensions of subshifts. A d-dimensional d subshift is a subset of AZ , where A is a finite set of symbols (finite alpha- bet). There is a natural action σ of Zd on this set, defined by n [σ (ω)]k = ωk−n, for n ∈ Zd. d Given ω ∈ AZ , the subshift associated with ω is the smallest subset of d AZ which contains ω, is closed and is shift-invariant. Equivalently, it is the closure of the orbit of ω for the product topology. The notions of aperiodicity and repetitivity are straightforward. Finite local complexity is automatic if A is finite.

Definition 2.15. — Let Ξ be an aperiodic, repetitive subshift on Zd. The suspension of Ξ is the following space: d SΞ = Ξ × R / ∼ where ∼ is the equivalence relation generated by n d (ω, x) ∼ (σ (ω), x − n), n ∈ Z . It carries naturally an Rd-action by translations. In practice, it is convenient to view elements of Ξ as tilings by d-cubes of edge-length 1, and (ω, x) ∈ SΞ can be interpreted as ω + x. The tiling topology, as deﬁned above, coincides with the quotient topology on this suspension. The following deﬁnition is standard.

Definition 2.16. — Two subshifts Ξ1 and Ξ2 are said to be ﬂow- equivalent if their suspensions are homeomorphic. The main results of this paper can therefore be interpreted in a symbolic setting: they measure how the complexity and repetitivity function of two subshifts relate when the subshifts are ﬂow equivalent.

3. Homeomorphisms between tiling spaces

3.1. Statement of the results

In this paper, the basic setting is to consider a homeomorphism between two aperiodic repetitive tiling spaces with ﬁnite local complexity. What can

ANNALES DE L’INSTITUT FOURIER COMPLEXITY AND HOMEOMORPHISMS BETWEEN TILING SPACES 555 be said about the underlying tilings if the two tiling spaces are homeomorphic? The first result is concerned with the complexity function of the tiling spaces: if two spaces are homeomorphic, the complexity functions are asymptotically equivalent in the sense of Definition 2.12. If the complexity is high, then the tiling systems may have entropy. In this case, the entropy is not preserved by homeomorphism (even though the fact that it is positive is). However, if the complexity grows polynomially, then the exponent is preserved by homeomorphism. Theorem 3.1. — Let h :Ω −→ Ω0 be a homeomorphism between two aperiodic repetitive tiling spaces with finite local complexity. Then there exist a transversal Ξ ⊂ Ω, a transversal Ξ0 ⊂ Ω0 and constants c, C, m, M > 0 such that for all r big enough,

(3.1) cpΞ0 (mr) 6 pΞ(r) 6 CpΞ0 (Mr). Corollary 3.2. — Let Ω be a (repetitive, aperiodic, FLC) tiling space, with polynomial complexity function, which means that there exists an α such that for some Ξ (and hence for all Ξ), there are two constants C1,C2 > 0 such that for all r big enough, α α C1r 6 pΞ(r) 6 C2r . Then the exponent α is a homeomorphism invariant of the space: any tiling space which is homeomorphic to Ω has polynomial complexity function with the same exponent α. In the language of symbolic dynamics, the results above can be stated as follows: If two minimal aperiodic Zd-subshifts are flow-equivalent, then their complexity functions are equivalent in the sense of Equation 3.1. If in addition they have polynomial complexity, then the exponents are the same. It is interesting to combine these results with some other results obtained previously on complexity. It was remarked in [12] that the complexity of a tiling can be related with the box-counting dimension of the transversals for the metric defined in Section 2.1. Proposition 3.3 ([12]).— Given a tiling space Ω and a canonical transversal Ξ, endowed with the metric defined before, the box-counting dimension of Ξ is given by the formula:

log(pΞ(r)) dimB(Ξ) = lim r→+∞ log(r) when it exists.

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Corollary 3.4. — The box-counting dimension of the transversals of FLC, aperiodic, repetitive tiling spaces is preserved by homeomorphism between the tiling spaces. It is striking to see that a homeomorphism between two tiling spaces preserves a quantity which is essentially metric. It is not the first time that links between topology and complexity are investigated. In [10], some relationships were established between the rank of the cohomology groups over the rationals on the one hand, and the asymptotic growth of the complexity function. For example in dimension 1, a tiling space with at most linear complexity has finitely generated Čech cohomology groups over the rationals. For cut-and-project tilings, the in- terplay is even richer: the complexity function is bounded above by rd (with d the dimension of the space) if and only if the rational cohomology groups are finitely generated. The results of this paper push further in this direc- tion: the complexity is not only loosely related with the topology of the space; it really is homeomorphism-invariant. The repetitivity function of a tiling is another quantity which is preserved by homeomorphism, as shown by the following result. Theorem 3.5. — Let h :Ω −→ Ω0 be a homeomorphism between two aperiodic repetitive tiling spaces with finite local complexity. Then there exist a transversal Ξ ⊂ Ω and a transversal Ξ0 ⊂ Ω0 and four constants c, C, m, M > 0 such that for all r big enough,

cRΞ0 (mr) 6 RΞ(r) 6 CRΞ0 (Mr). The study of the repetitivity function is a quantitative approach. It is somehow more common in the literature to make the qualitative distinction between linearly repetitive tiling spaces and the others. A tiling space is linearly repetitive if its repetitivity function grows no faster than r 7→ λr for some constant λ. Linearly repetitive tilings are amongst those with the highest degree of regularity. Many of the tilings used to model quasicrystals have this property, including the well-known Penrose tilings. This result then follows from the theorem. Corollary 3.6. — Let h :Ω −→ Ω0 be a homeomorphism as in the previous theorem. Then (Ω, Rd) is linearly repetitive if and only if (Ω0, Rd) is.

3.2. Homeomorphisms and tiling groupoids

Given a dynamical system, it is possible to deﬁne a groupoid which carries information on the space, the group, and the action of the latter

ANNALES DE L’INSTITUT FOURIER COMPLEXITY AND HOMEOMORPHISMS BETWEEN TILING SPACES 557 on the former. Groupoids also serve as a generalisation of group action, when there is no group. For example, a tiling space carries an action of Rd. In one dimension, this action of R restricts to an action of Z on a transversal Ξ (by the first return map). In higher dimension, there is in general no group action on the transversal; however, a groupoid can be defined as a replacement. While the groupoid point-of-view can be enlightening, it is not essential for most of the paper. Groupoids can be ignored until the Section6. The crucial elements of this section (which shouldn’t be skipped at first) are Definition 3.9 and Theorem 3.10 below.

Definition 3.7. — A groupoid G is defined as a small category with inverses. It consists of a set of base points G(0), and a set of arrows G between them. Given x ∈ G, it has a source and a range s(x) and r(x) in G(0) and the product xy is well defined if r(y) = s(x). For all p ∈ G(0) there is an element ep ∈ G which is neutral for multiplication (and p is (0) −1 identified with ep, so that G ⊂ G), and for all x there is x such that xx−1 = r(x), x−1x = s(x). Finally, the product is associative, whenever it is defined.

Definition 3.8. — A tiling space Ω has an associated groupoid, denoted Ω o Rd, which is deﬁned as the direct product of Ω and Rd, with range map, source map and product deﬁned as follows:

s(T, x) = T, r(T, x) = T − x, (T, x) · (T 0, x0) = (T 0, x + x0) if T 0 − x0 = T.

The topology on this groupoid is that induced by Ω × Rd. Consider a homeomorphism h :Ω −→ Ω0 between two aperiodic, FLC and minimal tiling spaces. By Theorem 2.6, the path-connected components in such tiling spaces are the orbits. Since a homeomorphism has to send path-connected component to path-connected component, we can deﬁne the following map.

Definition 3.9. — Let h :Ω −→ Ω0 be a homeomorphism. Deﬁne for d d all T ∈ Ω a function hT : R → R by

h(T − x) = h(T ) − hT (x).

It is well-deﬁned, by aperiodicity of the tilings with which we are dealing.

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Theorem 3.10. — Given a homeomorphism h :Ω −→ Ω0 between two aperiodic, minimal, FLC tiling spaces, the function

Ω × Rd −→ Rd (T, x) 7−→ hT (x) is continuous in two variables. Furthermore, for any fixed T , hT is a homeomorphism, and −1 −1 hT = h h(T ). Proof. — The first step of this proof is to show that for any given T ∈ Ω, the map x 7→ hT (x) is continuous. The map hT is defined as the bijection which makes the following diagram commute:

hT Rd / Rd

h Orb(T ) / Orb(h(T )) where Orb(T ) is the orbit of T in Ω under the action of Rd. The diﬃcult point is that the action maps Rd continuously and bijectively onto Orb(T ) via x 7→ T − x; however, for the relative topology induced by that of Ω, Orb(T ) is not homeomorphic to Rd, and not even locally homeomorphic. Indeed, the image of an open set is not open (in general) for the induced topology, and Orb(T ) is not locally connected. However, Theorem 2.6 gives information about the local structure of the leaves, which allows to prove that hT is a homeomorphism. Here, the sequential characterization of continuity is used: T and x being d ﬁxed, let (xn)n∈N be a sequence converging to x. Let γ : [0, 1] → R be the −n −(n+1) continuous path from x1 to x, such that γ restricted to [1 − 2 , 1 − 2 ] is the constant speed parametrization of the segment [xn+1, xn+2]. To prove continuity of hT at x, it is enough to prove that hT ◦ γ(t) tends to hT (x) as t tends to 1. Since h, γ and of the action of Rd on Ω are continuous, h(T − γ(t)) is close to h(T − x) provided t is close to 1. Assume t0 is close enough to 1 so that for all t > t0, h(T −γ(t)) is in a chart box of the form V 'ϕ B(0, ε)×X around h(T − x), and such that ϕ(h(T − x)) = (0, ξ). For t > t0, the path t 7→ ϕ◦h(T −γ(t)) is a continuous path in B(0, ε)×X, and therefore must be included in a path-connected component of the form B(0, ε) × {ξ0}, and therefore be of the form (η(t), ξ0). Since ϕ ◦ h(T − γ(t)) needs to tend to (ξ, 0), then ξ0 = ξ.

ANNALES DE L’INSTITUT FOURIER COMPLEXITY AND HOMEOMORPHISMS BETWEEN TILING SPACES 559

Finally,

ϕ ◦ h(T − γ(t)) = ϕ(h(T ) − hT ◦ γ(t)) ⊂ B(0, ε) × {ξ}, = (η(t), ξ).

Now, η(t) tends to zero on the one hand, and is equal to hT (γ(t)) − hT (x) on the other hand (because ϕ commutes with the action of Rd whenever it is deﬁned, see Theorem 2.6). Therefore, hT (γ(t)) tends to hT (x), and hT is continuous. The second part of the proof is to show that the function is continuous d of two variables. Let (T0, x0) ∈ Ω × R . What needs to be shown is d ∀ ε > 0, ∃ δ > 0, ∀ (T, x) ∈ Ω × R , kx − x0k < δ and D(T,T0) < δ ⇒ khT (x) − hT0 (x0)k < ε.

Let ε > 0. Let δ0 be such that whenever T1,T2 are δ0-close, then their images by h are ε/3-close (using uniform continuity of h). Finally, let −1 −1 δ = min (δ0 + kx0k + 2δ0) , δ0/2 .

Let now T ∈ Ω in a δ-neighbourhood of T0. Then, up to small translation −1 of combined size less than 2δ < δ0, the tilings T and T0 agree on B(0, δ0 + d kx0k + δ0). Then, for all y0 ∈ B(0, kxk) and all y ∈ R with ky − y0k < δ 6 δ0/2, the tilings T − y and T − y0 agree at least on B(0, δ0) up to a small (less than δ) translation. This last statement holds in particular for y = λx (λ ∈ [0, 1]). Let x ∈ B(x0, δ), and T ∈ Ω be δ-close to T0. Then, for all λ ∈ [0, 1], h(T − λx) and h(T0 − λx0) are ε/3-close. Assume (by contradiction), that khT (x) − hT0 (x0)k > ε. Since hT and hT0 are continuous, and khT (0) − hT0 (0)k = 0, one can use the intermediate value theorem: there exists a λ0 ∈ [0, 1] such that khT (λ0x) − hT0 (λ0x0)k = ε. One has

h(T − λ0x) = h(T ) − hT (λ0x)

h(T0 − λ0x0) = h(T0) − hT0 (λ0x0).

On the one hand, h(T ) and h(T0) match around the origin up to a translation smaller than ε/3. On the other hand, so do the left-hand sides of the equations above. Besides, hT (λ0x) and hT0 (λ0x0) diﬀer by exactly ε, which ε is much smaller than the radius of a tile. It is a contradiction.

Remark 3.11. — The map (T, x) 7→ hT (x) is actually a groupoid morphism (where the additive group Rd is seen as a groupoid with one base- point). It results from a direct check:

hT (x + y) = hT (x) + hT −x(y).

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This relation can also be called a cocycle condition for a reason which will be addressed in the last section of this paper.

Remark 3.12. — As a consequence of this result, the following map

Ω o Rd −→ Ω0 o Rd (T, x) 7−→ (h(T ), hT (x)) is an isomorphism of topological groupoids. Therefore, by a theorem of Renault [23], the associated cross product C∗-algebras C(Ω) o Rd and C(Ω0) o Rd are isomorphic. It appears that much of the dynamics is con- strained by the topology of the tiling spaces, because of the local product structure. If the topological spaces are homeomorphic, the homeomorphism is automatically an orbit equivalence, and the groupoids and associated C∗- algebras (which encode the dynamics) are isomorphic.

3.3. Strategy for the proof

Frank and Sadun [8] have proved that whenever two tiling spaces are topologically conjugate, their complexity functions are equivalent. Their result involves a diﬀerent complexity function than the one which is used here, as it can measure complexity of tiling spaces without FLC. Here is how their result would read in the framework of the present paper.

Theorem 3.13. — Let h :Ω −→ Ω0 be a topological conjugacy between two tiling spaces (i.e. a homeomorphism which commutes with the actions of Rd by translation). We assume that the tiling spaces are aperiodic, minimal, with FLC. Then, there are two canonical transversals Ξ ⊂ Ω and Ξ0 ⊂ Ω0, such that the associated complexity functions p and p0 satisfy

0 mp(r − a) 6 p (r) 6 Mp(r + b). for some constants a, b, m, M, and all r big enough.

Note that by Proposition 2.9, it is not important which transversals Ξ and Ξ0 are chosen, as any complexity functions would then satisfy this inequality. We include a sketch of the proof, with a simplifying assumption that h sends a vertical transversal to a vertical transversal. Even in this simpliﬁed setting, the proof contains important ideas which will be used in the proof of Theorem 3.1. It is pointed out afterwards how these ideas are to be generalised.

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Proof. — Let Ξ ⊂ Ω be any canonical transversal. It is assumed that Ξ0 := h(Ξ) is a canonical transversal itself (this is a simplifying assumption, as it shouldn’t be expected in general). The goal is then to compare the functions p and p0 associated with Ξ and Ξ0 respectively. Let ε < 2−1/2. By uniform continuity of h, there is a δ < 2−1/2 such that for all T,T 0 ∈ Ξ, if T and T 0 are at least δ-close, then h(T ) and h(T 0) are at least ε-close. In terms of patches, it means that if T and T 0 agree on B(0, δ−1), then their images agree on B(0, ε−1). In particular, h(T ) and h(T 0) have the same local conﬁguration at the origin. Now, ε and δ being ﬁxed, let r > 0, big enough so that it is big compared to ε−1 and δ−1. Assume T and T 0 agree on a ball of radius r + ε−1. Then d 0 it means that for all x ∈ R with kxk 6 r, T − x and T − x agree on B(0, δ−1). Using the fact that h is a conjugacy, it means that for all such x, h(T ) − x and h(T 0) − x agree on B(0, ε−1). Therefore, T and T 0 agree up to radius r + δ−1 implies that their images agree up to radius r + ε−1. Therefore,

−1 0 −1 p(r + δ ) > p (r + ε ).

−1 An exchange of h and h provides the other inequality.

This proof has three essential ingredients:

(1) The image of a transversal is a transversal (simplifying assumption); (2) The map h is uniformly continuous, and it allows to control how two tilings agree on a small neighbourhood of the origin when their preimages do; (3) The conjugacy sends orbits to orbits in a trivial way (given T , the map hT is the identity map), so that patches of tilings in Ω can be compared to patches of tilings in Ω0.

In the case when h is just a homeomorphism, point (2) is still satisfied. Point (1) is not satisfied, however it is very close to being satisfied: for all ε > 0, there are transversals Ξ ⊂ Ω and Ξ0 ⊂ Ω0 such that h(Ξ) and Ξ0 are within distance ε. This important approximation lemma is proved in the next section. The main difficulty is point (3). Given a ball of radius r in an orbit, what is the radius of its image by h? We need to control how hT (B(0, r)) grows with r. Lemma 5.1 and its corollary show that hT is not far from being Lipschitz.

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4. An approximation lemma

The following key lemma addresses one of the points outlined in the strategy of the proof: a homeomorphism between two tiling spaces is very close to sending a given canonical transversal to a canonical transversal. It is an interesting result in its own right. It should be compared with results obtained by Rand and Sadun [22]: in their paper, it is proved that any continuous map between FLC aperiodic repetitive tiling spaces can be approximated arbitrarily well by a local map (a local map has the partic- ularity that it sends any canonical transversal to a canonical transversal, i.e. vertical transversal to vertical transversal, see Deﬁnition 2.7). However, the approximating map in their case is not guaranteed to be a homeomorphism when the original map is. Note also that the map g in the proof of Lemma 5.1 is local in the sense of Rand–Sadun, but is a priori not within a small distance of h, and is not guaranteed to be a homeomorphism either.

Lemma 4.1. — Let h :Ω −→ Ω0 be a homeomorphism. Then for all ε > 0, there is a vertical transversal Ξ ⊂ Ω, a vertical transversal Ξ0 ⊂ Ω0, 0 and a homeomorphism hε :Ω −→ Ω such that 0 (1) hε(Ξ) = Ξ ; (2) h and hε are isotopic; (3) there is a continuous function s :Ω → Rd, the norm of which is uniformly bounded by ε and such that hε = h − s. Proof. — Let ε > 0, which is assumed to be small compared to the inner radius of the tiles. By uniform continuity of h, there is a δ > 0 such 0 0 that D(T,T ) 6 δ implies D(h(T ), h(T )) 6 ε. Then, there is a chart box 0 V0 ⊂ Ω , of diameter less than ε. Let U0 be a chart box in Ω such that −1 0 U0 ⊂ h (V0). Then we have the following picture for some δ 6 δ and 0 ε 6 ε. h U0 / V0

' '

0 ϕ 0 0 B(0, δ ) × X0 / B(0, ε ) × X0 where ϕ is induced by h (so that the diagram commutes). See also Fig- ure 4.1. When no confusion can arise, ϕ will not be used, and this map will just be noted h, possibly with the precision “in coordinates”. With a good choice of charts, we can assume that ϕ(0, ξ) = (0, ξ0) for 0 0 0 some ξ ∈ X0 and ξ ∈ X0. Call T0 and T0 the tilings corresponding to 0 0 (0, ξ) and (0, ξ ) respectively via the chart maps (so that h(T0) = T0). Note

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h(U0)

ϕ h h(T0) U0 T0 ≃ V0

Figure 4.1. U0 and V0, as initially built. They need pruning: for example V0 is too tall. First, pick a box-shaped neighbourhood of h(T0) 0 inside h(U0) (dark-grey neighbourhood V ). Then, widen it (light-grey neighbourhood V ). Finally, trim U0 so that its image is now included in the light-grey neighbourhood.

0 that h(U0) is open in V0. Therefore, there is a neighbourhood of T0, say 0 0 0 0 0 0 V ' B(0, ε1) × X such that V ⊂ h(U0). Let now V ' B(0, ε ) × X , and U be the largest open subset of U0 such that h(U) ⊂ V . It is easy to check 0 that it is of the form B(0, δ ) × X, with X a clopen subset of X0. Let Ξ and Ξ0 be the vertical transversals included in U and V respectively, such that Ξ '{0} × X and Ξ0 '{0} × X0. We now have U and V two chart domains with h(U) ⊂ V , and two transversals Ξ ⊂ U and Ξ0 ⊂ V such that Ξ0 ⊂ h(U).

X Ξ h(Ξ) X′ Ξ′ ≃ ≃

ϕ h U T0 ≃ h(T0) V

h(U)

Figure 4.2. The construction of Ξ and Ξ0. The| diameter{z } of U is at most δ and the diameter of V is at most ε.

The next step in the proof is to build a map Ξ → Ξ0 which is induced by h. The deﬁnition itself is straightforward: for each T ∈ Ξ, there is a unique T 0 ∈ Ξ0 which belongs to the same connected component of V as h(T ).

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0 Define hε(T ) to be this T . It is clear that for all T ∈ Ξ, hε(T ) = h(T )−s(T ), where s(T ) is a translation vector of norm less than ε (since the diameter of V is less than ε). Two things need to be shown: first, that hε is indeed continuous; then, that it can be extended to a function on all Ω and not just on Ξ. To this end, we first prove that s is a continuous function on Ξ. Let η > 0, which is assumed to be small with respect to the size of the tiles, and T1 ∈ Ξ. By continuity of h, there is ν > 0 such that for all T ∈ Ξ which is ν-close to T1, the images by h of T and T1 are η/2-close. It implies d that there is an x ∈ R , with kxk < η, such that h(T0) and h(T ) − x match exactly around the origin. In addition, since hε(T ) = h(T ) − s(T ) and hε(T0) = h(T0) − s(T0) both belong to the same canonical transversal Ξ0, they also need to match exactly around the origin. Therefore, when −1 restricted to B(0, ε ), one has (the notation =r stands for “agree around the origin up to radius r”):

h(T0) − s(T0) =1/ε h(T ) − s(T ) so

h(T0) − s(T0) =1/ε h(T ) − (x + s(T0)), and since kxk and s(T0) are both smaller than ε (which is small compared to the size of tiles), the only way that the last two tilings can match around the origin is if s(T ) = s(T0) + x, which implies ks(T0) − s(T )k < η. Since η could have been chosen arbitrarily small, it proves that s is continuous on Ξ. This last fact provides an instant proof of the continuity of hε on Ξ: d indeed, hε = h − s, both h and s are continuous, and R acts continuously on Ω by translations. Let us now show that hε can be extended on Ω. It is done by extending s. First, ϕ (induced by h) extends to a map deﬁned on the closure of its domain (by continuity of h on Ω). It is still noted ϕ:

ϕ : B(0, δ0) × X −→ B(0, ε0) × X0

Then, for all ξ ∈ X, there is a map ϕξ : x 7→ ϕ(x, ξ) deﬁned on the closed ball of radius δ0, which is a homeomorphism onto its image. By construction, −1 its image contains 0 in its interior. Let xξ := ϕξ (0). It is an inner point of the closed disk. Then, let us explicitly provide a homeomorphism fξ from the closed disk to itself, which leaves the boundary invariant and sends 0 to xξ.

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Let a(y) = 1 − kyk /δ0 deﬁned on B¯(0, δ0). Notice that it is 0 on the boundary, and takes value 1 at 0. Now, deﬁne

(4.1) fξ(y) = y + a(y)xξ.

It is clear that fξ(0) = xξ, and fξ leaves the boundary invariant. Further- 0 more, y 7→ a(y)xξ is k-Lipschitz, with k = kxξk /δ < 1. Therefore, fξ is one-to-one, and so it is a homeomorphism onto its image. Now, one can check that the norm of fξ(y) is 0 kfξ(y)k 6 kyk + kxξk − kyk kxξk /δ kyk kx k kyk kx k = δ0 + ξ − ξ δ0 δ0 δ0 δ0 and a simple study of the function (α, β) 7→ α+β−αβ on the domain [0, 1]2 shows that it is bounded by one. Therefore, the range of fξ is included in the closed ball B¯(0, δ0), and it contains its boundary. By an argument of (1) algebraic topology , fξ is therefore onto, and is a homeomorphism of the closed ball. It is a quick check that the function ξ 7→ fξ is continuous on Ξ. Indeed, −1 the continuity of hε implies that ξ 7→ (xξ, ξ) = h ◦hε(0, ξ) is a continuous function of ξ (expressed in coordinates, the chart maps being implicit). In addition, the map x 7→ (id(·) + a(·)x) is continuous for the topology of uniform convergence on the space of functions on the ball. It gives continuity of ξ 7→ fξ. 0 Now, it is possible to extend the function hε on U ' B(0, δ ) × X: in coordinates, it is defined as hε(y, ξ) = (fξ(y), ξ). It amounts to precom- posing h with fξ on each leaf. It is clear by definition of the fξ that this map restricts to hε on Ξ, as expected. The continuity of ξ 7→ fξ shows that hε is continuous on U. Finally, because of the boundary conditions put on fξ, the function hε can be extended continuously by h on the complement of U. The last point to prove is the isotopy between hε and h. It can be done by replacing the definition of fξ in Equation (4.1) by the following one:

fξ,t(y) = y + ta(y)xξ, for a parameter t ∈ [0, 1]. Leaving the rest of the construction unchanged, the result is a family of functions (htε)t∈[0,1]. For t = 1, the function is just hε; for t = 0, it is h0 = h. This provides the desired isotopy.

(1) Assume w in the interior of the ball is not on the range of fξ. Then fξ composed with z 7→ δ0(z − w)/ kx − wk maps the closed ball onto the closed sphere continuously, which is a contradiction.

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5. Proof of the theorems

5.1. Preliminary lemmas

0 The function hε being a well defined homeomorphism Ω → Ω , it is d d possible for all T ∈ Ω to define hε,T : R → R in the same way hT was defined.

Lemma 5.1. — There is an M > 0 such that for all T ∈ Ξ and all x ∈ Rd satisfying T − x ∈ Ξ,

khε,T (x)k 6 M kxk .

0 0 0 Proof. — Let T ∈ Ξ and T = hε(T ) ∈ Ξ . Let D and D be the pointing rules associated with transversal Ξ and Ξ0 respectively. It is immediate to 0 0 check that x ∈ D(T ) if and only if hε,T (x) ∈ D (T ). Let T be a triangulation rule locally derived from D, such that the set of vertices is exactly D. A triangulation is a tiling where each tile is a d-simplex, and tiles meet face-to-face (the faces being lower-dimensional simplices). One well known way to do so is to use Delaunay triangulations of Rd with vertices in D(T ). Such triangulations are dual to the Voronoi construction. In some (non-generic) cases, the Delaunay triangulation is not unique, but by making consistent choices, it is possible to ensure that T (T ) is locally derived from T . See [25], for a reference. Let g : Rd → Rd be the function deﬁned as follows: given x ∈ Rd, it belongs to a simplex, so it is uniquely written as a convex combination (with non-negative coeﬃcients) of its vertices, which are elements in D:

d X x = λ0v0 + λ1v1 + ... + λdvd, λk > 0, λk = 1. k=0 Then, deﬁne

d X g(x) = λkhε,T (vk). k=0 This map g is onto, but does not need to be one-to-one, see Figure 5.1. It is affine by parts, and using finite local complexity of T (T ), there are finitely may linear maps underlying the affine maps. A consequence of this is that g is Lipschitz, with coefficient M. Finally, g agrees with hε,T on D.

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Figure 5.1. A local pattern of the triangulation can be sent by g to a pattern which is combinatorially a triangulation, but with an overlap in the geometric realization. Remark that the shaded triangle has a “wrong” orientation on the right. In this case, the map g is not one- to-one.

Now, let us prove the lemma. Let x ∈ Rd such that T − x ∈ Ξ (that is: let x ∈ D). Then

khε,T (x)k = khε,T (x) − hε,T (0)k = kg(x) − g(0)k 6 M kx − 0k = M kxk .

This lemma has the following consequence.

Corollary 5.2. — Let h :Ω −→ Ω0 be a homeomorphism between two aperiodic repetitive FLC tiling spaces. There is an M > 0 and a C > 0 such that for all T ∈ Ω and all x ∈ Rd,

khT (x)k 6 M kxk + C.

Proof. — Let Ξ and Ξ0 be the vertical transversal involved in the construction of hε, and let D be the local pointing rule associated with Ξ. Using the Delaunay property of D(T ) for T ∈ Ω, there is R > 0 such that any ball B(x, R) intersects D(T ), for any T ∈ Ω. Using continuity of (T, x) 7→ hε,T (x) (see Theorem 3.10), the image of the compact set Ω × B(0,R) is compact, and in particular is bounded. It is therefore included in B(0, c) for some c.

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Let now be x ∈ Rd and T ∈ Ω. Then there are y, z ∈ Rd of length less than R such that T − y ∈ Ξ and T − x − z ∈ Ξ. One can write:

khε,T (x)k = khε,T (y + (x + z − y) − z)k

= khε,T (y) + hε,T −y(x + z − y) + hε,T −x−z(−z)k 6 c + M(kxk + kyk + kzk) + c 6 2c + 2MR + M kxk .

The equality from first line to second uses the definition of hε,T , the first inequality follows from Lemma 5.1. In the end, since khT (x) − hε,T (x)k 6 ε for all T and x, the result follows, with C = 2c + 2MR + ε.

These results are key for estimating the size of hT (B(0,R)) as R grows. A few additional lemmas still need to be proved before tackling the main theorem. The two following lemmas deal with the following problems: how can a given x ∈ Rd be approximated by a sum of small “return vectors” (Deﬁni- tion 5.4); and how hε,T (v) is locally constant in T when v is a small return vector.

Lemma 5.3. — Let D be a Delaunay set which is R-relatively dense, that is: [ d B(p, R) = R . p∈D d 0 Let x ∈ D and y ∈ R be such that kx − yk > 2R. Then there exists x ∈ D, such that

0 0 kx − xk 6 3R and kx − yk 6 kx − yk − R. Proof. — Deﬁne x − y z = x − 2R . kx − yk By the Delaunay property, there exists x0 ∈ D which is within distance R 0 of z. It is a quick check that kx − x k 6 3R, and 0 0 kx − yk 6 kx − zk + kz − yk 6 R + kx − yk − 2R = kx − yk − R. Definition 5.4. — Let Ω be a tiling space, and Ξ be a canonical transversal associated with the local pointing rule D. Then the set

d V = {v ∈ R ; ∃ T ∈ Ξ,T − v ∈ Ξ}

ANNALES DE L’INSTITUT FOURIER COMPLEXITY AND HOMEOMORPHISMS BETWEEN TILING SPACES 569 is discrete (by FLC), and is called the set of return vectors on Ξ. By deﬁ- nition, [ V = D(T ). T ∈Ξ Given v ∈ V, deﬁne

Ξv = {T ∈ Ξ; T − v ∈ Ξ}. By ﬁnite local complexity, it is a clopen subset of Ξ.

0 Lemma 5.5. — Let hε :Ω −→ Ω be a homeomorphism which sends a transversal Ξ to a transversal Ξ0. Let V be the set of return vectors on Ξ and, for λ > 0, let Vλ be those return vectors of length at most λ. Then there exists ν > 0 such that for all v ∈ Vλ and all T1,T2 ∈ Ξv,

D(T1,T2) 6 ν =⇒ hε,T1 (v) = hε,T2 (v).

Proof. — Given v ∈ Vλ, the map

T 7→ hε,T (v) 0 is continuous on Ξv, and it maps v to return vectors on Ξ (indeed, by definition of Ξv, both T and T − v are in Ξ, and therefore their images are in Ξ0). It is a continuous map from a compact set to a discrete set (by finite local complexity). It has therefore finite range, and is locally constant. This gives a constant νv, such that

D(T1,T2) 6 νv =⇒ hε,T1 (v) = hε,T2 (v).

Since Vλ is ﬁnite, the lemma follows.

5.2. Proof of Theorem 3.1 on complexity

0 0 Let Ξ, Ξ and hε be given by Lemma 4.1. Denote by D and D the local pointing rules associated with Ξ and Ξ0. The strategy of this proof is to use Lemma 5.1 to prove that whenever T,T 0 ∈ Ξ disagree within radius r, then their images in Ξ0 need to disagree within radius Mr, for some constant M which does not depend on r. It will prove that pΞ(r) 6 pΞ0 (Mr). The other inequality will be obtained by repeating the same argument for h−1. Let R be the constant such that the Delaunay sets D(T ) are R-relatively dense. Let ν be the constant given by Lemma 5.5 for λ = 3R, so that if

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−1 v ∈ V3R and T1,T2 ∈ Ξv agree up to radius ν , then hε,Ti (v) does not depend on i. Let M be as in Lemma 5.1. Let δ > 0, wich is assumed to be small with respect to the radius of the tiles, and smaller than ν. Assume also that −1 (2) −1 δ > 3R. By uniform continuity of hε , there is η > 0 such that −1 −1 ∀ T1,T2 ∈ Ξ,D(T1,T2) > (δ + 3R) ⇒ D(hε(T1), hε(T2)) > η.

Given r > 0, let T1,T2 ∈ Ξ be such that −1 −1 D(T1,T2) > (δ + r) , which means that these two tilings fail to agree on a ball of radius r + δ−1. d Then, there exists x ∈ R with kxk 6 r, such that T1 − x and T2 − x fail to agree on a ball of radius δ−1. The natural follow-up is to claim −1 that hε(T1) − hT1 (x) and hε(T2) − hT2 (x) disagree within radius η , and since khε,Ti (x)k 6 M kxk, then hε(T1) and hε(T2) disagree within radius Mr + η−1. This line of reasoning fails because there is no guarantee that hε,T1 (x) = hε,T2 (x). Some precautions have to be taken. Run the following algorithm, which is illustrated by Figure 5.2:

(1) Initialize k = 0, and start with xk = x0 = 0; −1 (2) If T1 − xk and T2 − xk do not agree on B(0, 3R + δ ), stop the algorithm. If they do agree (which means that kx − xkk must be at least 3R), continue to step (3). (3) Using Lemma 5.3, there exists xk+1 ∈ D(Ti) such that kxk −xk+1k 6 3R, and kx − xk+1k 6 kx − xkk − R. Since T1 − xk and T2 − xk agree at least up to radius 3R, the choice of xk+1 can be made the same for T1 and T2. (4) Return to step (2).

This algorithm terminates. Indeed, the distance between xk and x is reduced at each step by at least R. After ﬁnitely many steps, xk and x are within distance less than 3R of each other. When this happens, the stopping condition (2) applies, and the algorithm stops. The result of the algorithm is a ﬁnite sequence of vectors (xi)i=0,...,k such that each vi := xi+1 − xi is a return vector to Ξ in V3R. Furthermore, for all i < k, T1 − xi and T2 − xi agree up to radius at least δ−1, which is greater than ν−1. Therefore, by Lemma 5.5,

hε,T1−xi (vi) = hε,T2−xi (vi), 0 and these are return vectors to Ξ , which we denote (wi)i=0,...,k−1.

(2) In its contrapositive form.

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In ﬁnitely many steps, xk xi comes within range 3R of x.

x vi

... 1 vk δ− x0 1 3R + δ−

T1 and T2 don’t agree on this ball.

The algorithm stops whenever T1 and T2 don’t agree on this ball. If they do agree, the next vi is a return vector for both T1and T2.

Figure 5.2. An illustration of the algorithm.

Because of the stopping condition of the algorithm above,

−1 −1 D(T1 − xk,T2 − xk) > (3R + δ ) , and therefore D(hε(T1 − xk), hε(T2 − xk)) > η. Now, just remark that

hε(Ti − xk) = h(Ti − v0 − v1 − ... − vk−1)

= hε(Ti) − w0 − ... − wk−1.

In particular, there is a same y such that hε(Ti − xk) = hε(Ti) − y, i = 1, 2. It was proved that hε(T1)−y and hε(T2)−y disagree on the ball of radius η−1. Additionally,

kyk 6 M kxkk 6 M(kxk + 3R), where the ﬁrst inequality is obtained by Lemma 5.1, and the second by the fact that kx − xkk 6 3R. To sum up, it was proved that whenever two tilings T1,T2 ∈ Ξ disagree within radius δ−1 + r, then their images have to disagree within radius Mr + C for a constant C = η−1 + 3MR. Therefore,

−1 pΞ(r + δ ) 6 pΞ0 (Mr + C).

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Up to changing M and restricting r to large values, one gets pΞ(r) 6 pΞ0 (Mr). The other inequality is obtained by reversing the roles of h and h−1.

5.3. Proof of Theorem 3.5 on repetitivity

The proof of this theorem is built on top of the proof of Theorem 3.1. The additional arguments are fairly simple, since repetitivity is very well translated in terms of recurrence and return vectors. How the norm of return vectors is changed by a homeomorphism h, in turn, is well evaluated by Lemma 5.1. 0 Let h :Ω −→ Ω be a homeomorphism and hε be the approximation obtained from Lemma 4.1, which sends the canonical transversal Ξ to the canonical transversal Ξ0. Let R and R0 be the respective repetitivity functions of Ω and Ω0 relative to Ξ and Ξ0. Let also M be the constant given by Lemma 5.1. The goal is to compute bounds on R(r) in terms of R0. Let P be a patch of radius r of the form T0 ∩ B(0, r) for T0 ∈ Ξ. It deﬁnes a clopen set of Ξ by

UP := {T ∈ Ξ; T ∩ B(0, r) = P }. Given r, we want to compute an upper bound for the the norm of a return vector from any tiling T ∈ Ξ to UP , in a way which is uniform in P . Let V := hε(UP ). It is a clopen set, since hε is a homeomorphism. Now, 0 0 consider the partition of Ξ : {VP 0 }P 0 , where P ranges over all patches of the form P 0 = T 0 ∩ B(0, Mr) with T 0 ∈ Ξ0. It is claimed that any element of this partition is either completely included in V or disjoint. Indeed, let 0 0 0 T1,T2 ∈ Ξ be two tilings which agree up to radius Mr (so that they belong to the same element of the partition defined above). It was proved in the proof of Theorem 3.1 that whenever T1,T2 ∈ Ξ disagree within radius r, then their images need to disagree within radius Mr. In a contrapositive −1 0 −1 0 form, it shows that hε (T1) and hε (T2) agree up to radius r. So they are 0 0 either both in UP , or none is. So either T1 and T2 are both in V , or none is. It results that k G 0 0 0 0 h (U ) = V 0 , where P = T ∩ B(0, Mr) for some T ∈ Ξ . ε P Pi i i i i=1 0 To conclude, given T ∈ Ξ let us find a return vector to UP . Let T := hε(T ). 0 By definition, there is y be such that T − y ∈ V 0 and kyk R(Mr) + 1. Pi 6

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−1 So x := (hε )T 0 satisﬁes T − x ∈ UP . We use the Lipschitz-like estimation −1 of Lemma 5.1 on (hε )T 0 to get: ˜ 0 x 6 M(R (Mr) + 1) + C.

For any T ∈ Ξ it is possible to obtain a return vector to UP in this way. It shows (up to renaming the constants and maybe restricting to r big enough) that 0 R(r) 6 λR (Mr). The other inequality, again, is obtained by reversing the roles of h and h−1.

6. Concluding remarks on deformations

There have been essentially two approaches for understanding tiling deformations. In the first one (due to Clark and Sadun [7]) a deformation is defined as a map from the set of edges of a polytopal tiling to Rd. The image of an edge is a (a priori) different edge; fitting edges together, it defines a deformation of the tiling. This map needs to be pattern-equivariant in the sense that the image of an edge of T should be determined by the local pattern around it. The other point of view uses Kellendonk’s pattern-equivariant differential forms [14, 15]. A function defined on Rd is T -equivariant for some fixed tiling T if its value at x only depends on the local pattern of T around x. A deformation of a point pattern D(T ) is then the image f(D(T )) under a differentiable function f : Rd → Rd, with T -equivariant differential. In both cases, a deformation is associated with an element in the cohomology of the tiling space with coefficients in Rd: the Čech cohomology on the one hand (interpreted in [7] by using the pattern-equivariant formalism), and the pattern-equivariant De Rham cohomology on the other hand. These cohomology groups turn out to agree [6,7, 16]. In the work of Clark and Sadun, a deformation of an aperiodic, repetitive, FLC tiling of Rd by polytopes T is given by a map f defined on the set of oriented edges of T , valued in Rd, such that for some R > 0, (1) f(e) only depends on the local configuration of T around e up to radius R; Pn (2) if e1, . . . , en is a closed circuit of edges in T , then i=1 f(ei) = 0. The second condition is a cocycle condition. Such a function f is a coboundary if there is s defined on vertices of T , such that (1) s(v) only depends on (T − v) ∩ B(0,R);

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(2) f(e) = s(e+)−s(e−), where e+ and e− denote the target and source vertices of the oriented edge e. Then they prove that the quotient of cocycles by coboundaries is exactly isomorphic to Hˇ 1(Ω; Rd). Furthermore, such a cocycle defines a deformation of the tiling as follows: assuming (without loss of generality) that 0 is a vertex of T , define X F (v) = f(ei), i where (ei)i is a path of edges from 0 to the vertex v. By definition of f, the value of F (v) does not depend on the path chosen. Then, at least if f is close enough to the identity, F (T ) defines a tiling of Rd (a sufficient condition is that the image of a tile should be a non-degenerate tile with the same orientation). Furthermore, there is then a homeomorphism between the tiling spaces of T and F (T ). In light of this presentation, it appears that one can associate a deformation to any homeomorphism between two tiling spaces. Let h :Ω → Ω0 0 be a homeomorphism, and hε, Ξ, Ξ be the data provided by Lemma 4.1. Let D and D0 be the local pointing rules associated with Ξ and Ξ0, and let T0 ∈ Ξ. Let T be the local derivation introduced in the proof of Lemma 5.1, such that for any T ∈ Ξ, T (T ) is a triangulation with vertices D(T ). Let T 0 be the image rule, with vertices given by the pointing rule D0, and such that two vertices are linked by an edge in T 0 if and only if their preimages by hε are linked by an edge in T . Even though the image of the triangulation T does not need to be a triangulation itself (see Figure 5.1), the map 0 T (T ) 7→ T (hε(T )) is induced by a deformation in the sense above. Let us make it explicit: let e0 be an edge of the triangulation T (T0). Say e0 = (v, e) where v is a vertex of T (T0) (the source of e0), and e is a vector d of R . Define Ξe := Ξ ∩ (Ξ − e) (the sub-transversal of all tilings in Ξ which have an edge e at the origin, pointing away). Then the map T 7→ hε,T (e) is continuous, and its image is a return vector to Ξ0. By Lemma 5.5, the image of e under hε,T only depends on the configuration of T around the origin up to a certain radius, say Re. Then, define f(e0) = hε,T0−v(e). Defining f similarly for other edges, and letting R = max{Re}, f defines a deformation cocycle with pattern-equivariance radius R. Then f induces 0 a deformation F : T (T0) 7→ T (hε(T0)). This function F extends to Ξ, and it coincides with hε on this set. It may be enlightening to adopt a groupoid point-of-view on these deformations. Let G be the reduction on Ξ of Ω o Rd; it consists of the elements with both range and source in Ξ. Elements of G are of the form (T, v)

ANNALES DE L’INSTITUT FOURIER COMPLEXITY AND HOMEOMORPHISMS BETWEEN TILING SPACES 575 where T ∈ Ξ and v is a return vector of T on Ξ. Then the deformation data in Clark–Sadun’s presentation is a partially defined map f : G → Rd. It is defined only on the elements (T, v) where v is a return vectors corresponding to an edge in T . That being said, those elements generate the groupoid and f satisfies the cocycle condition. Therefore, it extends to a groupoid morphism G → Rd. There is an equivalence between deformations in the sense of Clark–Sadun and groupoid morphisms G → Rd such that (T, v) 7→ f(T, v) is locally constant in the variable T ∈ Ξ. It is now very apparent that hε induces a deformation, since it was proved that (T, x) 7→ hε,T (x) is locally constant in T , if one restricts to T ∈ Ξ. ˇ 1 d The function F = (hε)|Ξ defines a class in H (Ω; R ). This cohomology group is isomorphic to Kellendonk’s strong pattern-equivariant cohomol- 1 d ogy group Hpe,s(Ω; R ). For another point-of-view, the groupoid morphism 1 d ˇ 1 d (T, x) 7→ hε,T (x) defines an element in H (G; R ) ' H (Ω; R ). It is eas- ily seen that two different approximations hε and hδ define two cohomology class which differ by a weak coboundary, i.e. by the coboundary of a groupoid 0-cocycle s :Ξ → Rd which is just continuous and not locally constant. It therefore appears that h defines a class in the strong cohomology group, which is well defined only up to a weak coboundary. It is the definition of Kellendonk’s mixed group. Theorem 6.1. — Let h :Ω −→ Ω0 be a homeomorphism between two aperiodic, FLC, repetitive tiling spaces. Then h defines an element in Kel- 1 d lendonk’s mixed cohomology group Hpe,m(Ω; R ). This element is the image in the mixed group of the deformation cocycle defined above under the quotient map 1 d 1 d Hpe,s(Ω; R ) −→ Hpe,m(Ω; R ) (see [15]). This element in the quotient does not depend on the choice of 0 Ξ, Ξ or hε. 0 00 Besides, if h1 :Ω → Ω and h2 :Ω → Ω are homeomorphisms which define the same element in the mixed group, then Ω0 and Ω00 are topologically conjugate. The second part of this theorem is proved in a very similar way as [7, 0 0 −1 0 Theorem 2.2]: for a fixed T , one defines ϕ(T ) − x := h2 ◦ h1 (T ) − x and show that it extends to a map between the tiling spaces. We omit the details. A refined version of this theorem, as well as a more complete discussion of homeomorphisms and orbit-equivalences between tiling spaces can be found in a later paper [11]. 1 d This theorem indicates that Kellendonk’s mixed group Hpe,m(Ω; R ) is an appropriate invariant to classify homeomorphisms of Ω to another FLC

TOME 67 (2017), FASCICULE 2 576 Antoine JULIEN tiling space modulo conjugacy. This should be compared to recent work of Kellendonk and Sadun [17] in which they show that some elements of the 1 group Hpe,s (the so-called “inﬁnitesimals”) classify topological conjugacy of FLC tiling spaces modulo invertible local derivation. Finally, it is legitimate to wonder whether a homeomorphism between tiling spaces can be perturbed in order to preserve all transversals (or equivalently, to choose Ξ in such a way that the map g in Lemma 5.1 is a homeomorphism). Conjecture. — Given a homeomorphism h :Ω → Ω0 between two aperiodic repetitive FLC tiling spaces, there is an homeomorphism h0, isotopic to h, such that for any canonical transversal Ξ in Ω, h0(Ξ) is a canonical transversal in Ω0. A positive answer was given in [11] when Ω is uniquely ergodic. However, it would be surprising if this assumption couldn’t be dropped.

BIBLIOGRAPHY

[1] M. Baake, D. Lenz & C. Richard, “Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies”, Lett. Math. Phys. 82 (2007), no. 1, p. 61-77. [2] M. Baake, M. Schlottmann & P. D. Jarvis, “Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability”, J. Phys. A 24 (1991), no. 19, p. 4637-4654. [3] J. Bellissard, R. Benedetti & J.-M. Gambaudo, “Spaces of tilings, finite tele- scopic approximations and gap-labeling”, Comm. Math. Phys. 261 (2006), no. 1, p. 1-41. [4] J. Bellissard, D. J. L. Herrmann & M. Zarrouati, “Hulls of aperiodic solids and gap labeling theorems”, in Directions in mathematical quasicrystals, CRM Monogr. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2000, p. 207-258. [5] R. Benedetti & J.-M. Gambaudo, “On the dynamics of G-solenoids. Applications to Delone sets”, Ergodic Theory Dynam. Systems 23 (2003), no. 3, p. 673-691. [6] H. Boulmezaoud & J. Kellendonk, “Comparing different versions of tiling cohomology”, Topology Appl. 157 (2010), no. 14, p. 2225-2239. [7] A. Clark & L. Sadun, “When shape matters: deformations of tiling spaces”, Er- godic Theory Dynam. Systems 26 (2006), no. 1, p. 69-86. [8] N. Frank & L. Sadun, “Fusion tilings with infinite local complexity”, Topology Proc. 43 (2014), p. 235-276. [9] A. Julien, “Complexité des pavages apériodiques : calculs et interprétations”, PhD Thesis, Université Lyon 1, 2009, https://tel.archives-ouvertes.fr/ tel-00466323v1. [10] ——— , “Complexity and cohomology for cut-and-projection tilings”, Ergodic The- ory Dynam. Systems 30 (2010), no. 2, p. 489-523. [11] A. Julien & L. Sadun, “Tiling deformations, cohomology, and orbit equivalence of tiling spaces”, http://arxiv.org/abs/1506.02694, 2015. [12] A. Julien & J. Savinien, “Transverse Laplacians for substitution tilings”, Comm. Math. Phys. 301 (2011), no. 2, p. 285-318.

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[13] J. Kellendonk, “Noncommutative geometry of tilings and gap labelling”, Rev. Math. Phys. 7 (1995), no. 7, p. 1133-1180. [14] ——— , “Pattern-equivariant functions and cohomology”, J. Phys. A 36 (2003), no. 21, p. 5765-5772. [15] ——— , “Pattern equivariant functions, deformations and equivalence of tiling spaces”, Ergodic Theory Dynam. Systems 28 (2008), no. 4, p. 1153-1176. n [16] J. Kellendonk & I. F. Putnam, “The Ruelle-Sullivan map for actions of R ”, Math. Ann. 334 (2006), no. 3, p. 693-711. [17] J. Kellendonk & L. Sadun, “Meyer sets, topological eigenvalues, and Cantor ﬁber bundles”, J. Lond. Math. Soc. 89 (2014), no. 1, p. 114-130. [18] J. C. Lagarias & P. A. B. Pleasants, “Repetitive Delone sets and quasicrystals”, Ergodic Theory Dynam. Systems 23 (2003), no. 3, p. 831-867. [19] D. Lenz, “Aperiodic Linearly Repetitive Delone Sets Are Densely Repetitive”, Dis- crete Comput Geom 31 (2004), no. 2, p. 323-326 (en). [20] M. Morse & G. A. Hedlund, “Symbolic Dynamics”, Amer. J. Math. 60 (1938), no. 4, p. 815-866. [21] P. S. Muhly, J. N. Renault & D. P. Williams, “Equivalence and isomorphism for groupoid C∗-algebras”, J. Operator Theory 17 (1987), no. 1, p. 3-22. [22] B. Rand & L. Sadun, “An approximation theorem for maps between tiling spaces”, Discrete Contin. Dyn. Syst. 29 (2011), no. 1, p. 323-326. [23] J. Renault, A groupoid approach to C∗-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980, ii+160 pages. [24] L. Sadun & R. F. Williams, “Tiling spaces are Cantor set ﬁber bundles”, Ergodic Theory Dynam. Systems 23 (2003), no. 1, p. 307-316. [25] M. Schlottmann, “Periodic and quasi-periodic Laguerre tilings”, Internat. J. Mod- ern Phys. B 7 (1993), no. 6-7, p. 1351-1363.

Manuscrit reçu le 23 juillet 2015, révisé le 14 juin 2016, accepté le 12 juillet 2016.

Antoine JULIEN Institutt for Matematiske fag NTNU 7491 Trondheim (Norway) Current address: Nord universitet 7600 Levanger (Norway) [email protected]

TOME 67 (2017), FASCICULE 2